TL;DR
Terence Tao reports that an instance of Erdős Problem #728 was resolved largely by AI tools after iterative feedback and reconstruction of the intended question. The work combined large-language-model drafts, formalization in Lean, automated repairs, and multiple rounds of AI-driven rewriting to produce a Lean-verified proof and several human-readable expositions.
What happened
Erdős Problem #728, an unsettled question about factorial divisibility arising from work in the 1970s, was recently attacked by a sequence of AI tools and community participants. An initial run by a team using the tool AlphaProof found trivial solutions when parameters were allowed to be large, prompting the community to add the constraint a,b ≤ (1−ε)n to match the intended spirit of the question. On Jan. 4, ChatGPT produced a proof for the adjusted problem with a small constant C; that draft was formalized into the Lean proof assistant by an AI system called Aristotle. Community participants then used ChatGPT to translate the Lean formalization into natural-language drafts, iteratively filling gaps. ChatGPT later adapted the argument to handle a large C, and Aristotle automatically repaired remaining errors to yield a Lean-verified proof. Multiple rounds of AI editing produced increasingly polished writeups; Terence Tao noted the result appears not to have been replicated in existing literature, though similar statements and methods were known.
Why it matters
- Demonstrates recent AI capability to generate and refine mathematical arguments and expositions rapidly.
- Shows how large-language models and formal proof assistants (Lean) can be used together to produce machine-checked proofs.
- Highlights a new workflow where AI can produce multiple alternate writeups and assist in shortening or repairing proofs.
- Raises questions about assessing novelty and overlap with existing mathematical literature.
Key facts
- The subject is Erdős Problem #728, about divisibility relations among factorials and binomial-coefficient factorizations.
- The problem had been ambiguously worded; community reconstruction added the constraint a,b ≤ (1−ε)n to avoid trivial large-parameter solutions.
- An AlphaProof-associated team found trivial solutions under the original loose interpretation, prompting the tighter constraint.
- On Jan. 4, ChatGPT produced a proof for the constrained problem with a small constant C; that version was formalized into Lean by Aristotle.
- ChatGPT was used to translate Lean formalizations into readable drafts; subsequent AI-assisted edits filled gaps and improved exposition.
- ChatGPT later adapted the argument to handle a larger constant C; Aristotle repaired minor errors and produced a Lean-verified proof.
- Terence Tao stated the result appears, to the best of their knowledge, not to be present exactly in existing literature, though related results exist.
- Tao remarked on the notable capability of AI to rapidly write, rewrite, and tailor expositions, while preferring a primary human-authored final paper.
What to watch next
- Whether the community will produce a single, human-authored canonical paper tied to the AI-assisted Lean verification (not confirmed in the source).
- Efforts to develop objective measures to compare AI-produced proofs to existing literature or to quantify novelty (discussed but not confirmed in the source).
- Broader adoption of combined LLM-plus-formal-verification workflows in mathematical research and publishing (not confirmed in the source).
Quick glossary
- Erdős problem: A research question originating from the work or posed by mathematician Paul Erdős, often catalogued for communal study.
- Lean: A formal proof assistant that can express mathematical definitions and verify proofs in a machine-checkable way.
- Formal verification: The process of using formal methods and proof assistants to check that a mathematical argument meets precise logical rules.
- Large language model (LLM): A class of machine-learning systems trained on large text corpora to generate human-like text and assist with reasoning and drafting.
- Binomial coefficient: A combinatorial quantity often written C(n,k) or (n choose k), equal to n!/(k!(n−k)!), commonly appearing in number-theoretic questions.
Reader FAQ
Was the proof produced entirely by AI without human intervention?
No. The process involved multiple AI tools and human participants; Tao describes the solution as produced 'more or less autonomously' after some feedback and community reconstruction.
Is this mathematical result new?
Tao states that, to the best of their knowledge, the specific result is not replicated in existing literature, though similar results and methods were located.
Were formal proof tools used to check the result?
Yes. An AI tool named Aristotle formalized the argument in Lean and repaired gaps to yield a Lean-verified proof.
Will AI replace human-written mathematical papers?
Tao expressed a preference for primary papers to remain human-authored in essential parts but acknowledged a role for AI in routine proofs and producing alternate expositions.
Back Terence Tao @tao@mathstodon.xyz Recently, the application of AI tools to Erdos problems passed a milestone: an Erdos problem (#728 https://www. erdosproblems.com/728 ) was solved more or less autonomously by…
Sources
Related posts
- Revealed: AI Startup Hark Hired Apple’s iPhone Air Designer in January
- AI Solves Erdős Problem #728, Terence Tao Reports on Mathstodon
- Convert a single photo into a navigable 3D Gaussian Splat with depth